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Arc length s of a logarithmic spiral as a function of its parameter θ. Arc length is the distance between two ... The length of the curve is given by the formula ...
The curve is thus parametrized in a preferred manner by its arc length. With a non-degenerate curve r(s), parameterized by its arc length, it is now possible to define the Frenet–Serret frame (or TNB frame): The tangent unit vector T is defined as :=.
Animation of the torsion and the corresponding rotation of the binormal vector. Let r be a space curve parametrized by arc length s and with the unit tangent vector T.If the curvature κ of r at a certain point is not zero then the principal normal vector and the binormal vector at that point are the unit vectors
The length l of a parametric C 1-curve : [,] is defined as = ‖ ′ ‖. The length of a parametric curve is invariant under reparametrization and is therefore a differential-geometric property of the parametric curve.
Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures ...
As a curve undergoes the curve-shortening flow, its length L decreases at a rate given by the formula =, where the integral is taken over the curve, κ is the curvature, and s is arc length along the curve. The integrand is always non-negative, and for any smooth closed curve there exist arcs within which it is strictly positive, so the length ...
Important quantities in the Whewell equation. The Whewell equation of a plane curve is an equation that relates the tangential angle (φ) with arc length (s), where the tangential angle is the angle between the tangent to the curve at some point and the x-axis, and the arc length is the distance along the curve from a fixed point.
In geometry, the Cesàro equation of a plane curve is an equation relating the curvature (κ) at a point of the curve to the arc length (s) from the start of the curve to the given point. It may also be given as an equation relating the radius of curvature (R) to arc length. (These are equivalent because R = 1 / κ .)